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(C) Let $z = \frac{\beta - \alpha}{1 - \overline{\alpha}\beta}$.We calculate $|z|^2 = z \cdot \overline{z} = \left( \frac{\beta - \alpha}{1 - \overlin

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(C) Let $z = \frac{\beta - \alpha}{1 - \overline{\alpha}\beta}$.We calculate $|z|^2 = z \cdot \overline{z} = \left( \frac{\beta - \alpha}{1 - \overline{\alpha}\beta} \right) \left( \frac{\overline{\beta} - \overline{\alpha}}{1 - \alpha\overline{\beta}} \right)$.Expanding the numerator: $(\beta - \alpha)(\overline{\beta} - \overline{\alpha}) = \beta\overline{\beta} - \beta\overline{\alpha} - \alpha\overline{\beta} + \alpha\overline{\alpha} = |\beta|^2 - \beta\overline{\alpha} - \alpha\overline{\beta} + |\alpha|^2$.Expanding the denominator: $(1 - \overline{\alpha}\beta)(1 - \alpha\overline{\beta}) = 1 - \alpha\overline{\beta} - \overline{\alpha}\beta + \alpha\overline{\alpha}\beta\overline{\beta} = 1 - \alpha\overline{\beta} - \overline{\alpha}\beta + |\alpha|^2|\beta|^2$.Since $|\beta| = 1$,we have $|\beta|^2 = 1$.Substituting $|\beta|^2 = 1$ into the expressions:Numerator: $1 - \beta\overline{\alpha} - \alpha\overline{\beta} + |\alpha|^2$.Denominator: $1 - \alpha\overline{\beta} - \overline{\alpha}\beta + |\alpha|^2(1) = 1 - \alpha\overline{\beta} - \overline{\alpha}\beta + |\alpha|^2$.Since the numerator and denominator are equal,$|z|^2 = 1$,which implies $|z| = 1$.

If $\alpha$ and $\beta$ are different complex numbers with $|\beta| = 1$,then $\left| \frac{\beta - \alpha}{1 - \overline{\alpha}\beta} \right|$ is equal to

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